PHY.4:5 - Archetypal Grounding

These are constructed physical accounts. They illustrate how physical premises constrain an unknown law and how a changed premise changes the result.

PHY.4:5.1 - Restrict an unknown resistive force

Seek the instantaneous mean resistive force on a body moving through a homogeneous isotropic medium at a fixed material state, in a classical three-dimensional regime. Assume the body and preparation introduce no preferred direction, relevant memory is negligible, and relative velocity w is the only varying input. These assumptions define the proposed comparison; their adequacy in a particular experiment remains a physical question.

Rotating the entire relevant situation rotates w and the force together. Because there is no further directional input, F(Qw)=QF(w) for every spatial rotation Q. For nonzero w, rotations about w rule out a perpendicular force component. Equal-length velocities are related by rotation, so write:

F(w)=-a(|w|^2)*w for w different from zero.

At w=0, rotational symmetry gives F(0)=0. The coefficient a is an unknown scalar function at the fixed material conditions. Its dimension is mass divided by time. Memoryless passive resistance gives:

F(w) dot w=-a(|w|^2)*|w|^2 <= 0,

so a(s)>=0 for s>0.

The result determines a direction and sign while leaving the speed dependence unresolved. Both a(s)=a0 and a(s)=b0*sqrt(s) with suitable nonnegative dimensional constants satisfy these restrictions. They predict different force magnitudes when speed changes. The comparison alone therefore provides neither a linear nor a quadratic drag law.

For a direction-only question, use the result directly. To predict stopping time, supply further information about a over the speeds involved, or obtain a sufficient bound over the admissible family. MMP.11 keeps that unresolved relation visible instead of inserting a familiar drag coefficient.

PHY.4:5.2 - Add a physical direction

Now the body is oriented or the environment has an aligned surface. Include its unit direction n in the input. Under a rotation of the complete situation, both w and n change, so the condition becomes:

F(Qw,Qn)=QF(w,n).

A rotation fixing w can now change n. The earlier argument that fixed all inputs while rotating a perpendicular output component is no longer available.

For example, one admissible linear resistive family is:

F(w,n)=-alpha*w-beta*(n dot w)*n.

For unit n, its power is -alpha*|w|^2-beta*(n dot w)^2. Resolving w into parts parallel and perpendicular to n shows passivity for every w when alpha>=0 and alpha+beta>=0. This is a possible family, not an exhaustive determination of the changed law.

With consistent units, take alpha=1, beta=3, w=(1,1,0) and n=(1,0,0). The force is (-4,-1,0) and power is -5. The force resists motion while pointing in a direction different from -w. This possibility is excluded by the earlier isotropic account and allowed by the additional physical input.

If n is fixed in the laboratory while only the body motion changes, retain that fixed n in the experiment’s account. Rotating the coordinate system changes the components of both quantities; it supplies no premise that removes the material or environmental direction.

PHY.4:5.3 - Exchange two participants

Two identical chambers at the same temperature exchange one kind of particle through a symmetric passage. Let a and b be their current concentrations and let J(a,b) be the instantaneous mean transfer rate from the first chamber to the second. Assume the proposed regime needs no additional passage state, and the surroundings introduce no directional bias.

Keep the chamber labels and the positive counting direction fixed, and exchange the concentrations in the two preparations. Because the chambers and passage are symmetric and the surroundings supply no bias, this physical exchange predicts a reversed measured transfer rate:

J(b,a)=-J(a,b).

At equal concentrations, J(a,a)=0. A proposed law J(a,b)=k*(a-b)^2 with k>0 fails the interchange condition: it gives a positive rate in the same counted direction after the preparations are exchanged.

Both J(a,b)=k1*(a-b) and J(a,b)=k3*(a-b)^3 satisfy the interchange condition when their constants have the corresponding units. Symmetry has not chosen between them or determined their coefficients. Directional thermodynamic claims would additionally use the physical driving potentials and the applicable dissipation law.

If the passage stores no particles, the chamber particle numbers satisfy dN1/dt=-J and dN2/dt=J, preserving their sum. If appreciable particles accumulate in the passage, include its particle number and separate inlet and outlet rates. The earlier two-chamber balance then omits a relevant participant.

This comparison uses exchange and balance rather than rotation. It opens the same kind of result: an admissible law family and the physical condition that would require revising it.